Let $A$, $B$ and $C$ be three sets. If $A$ belongs to $B$ and $B$ is a subset of $C$, is it true that $A$ is a subset of $C$ too?
The answer in my textbook reads –
No. Let $A=\{1\}$, $B=\{\{1\},2\}$ and $C=\{\{1\},2,3\}$. Here $A$ belongs to $B$ as $A=\{1\}$ and $B$ is a subset of $C$. But $A$ is not a subset of $C$ as $1$ belongs to $A$ and $1$ doesn't belong to $C$. Note that an element of a set can never be a subset of itself.
I am confused by the last note. Is the above mentioned explanation correct? Can someone explain in a better way?
Best Answer
A= {1} and since A belongs to B, it means set B consists of set and number.i.e. B contains a SET (set A) and a number (2). and since B is the subset of C all the element of B are in C. Therefore C contains a Set {1} 2 number '2' and '3'.
Since the definition of a subset is that every element of set A must be present in C. Therefore element in set A i.e. 1 is not present in set C.
p.s. Don't confuse a set with an element.